Proposition: Propagation of the layer limit

After the fixed spatial localization, for every integer $r\ge 0$,

Every spacetime derivative converges at rate $O(\epsilon )$ on fixed compact subsets of $(0,T_{\ast }]\times {\mathbb{R}}^3$.

Proof. For $t\ge \epsilon$, the ramp is identically one. Set

The exit expansions and smoothness of the right-hand solutions give $O(\epsilon )$ position and velocity data for both $z_j$ at $t=\epsilon$, in every spatial Sobolev order.

The first difference satisfies

For a free wave with initial time $s$, Fourier multipliers give, on a fixed bounded time interval,

Indeed, $\mathsf{C}$ is bounded in each Sobolev norm, $\mathsf{S}:H^r\to H^{r+1}$ is uniformly bounded on bounded time intervals, and ${\mathsf{C}}^{\prime }:H^{r+1}\to H^r$ is uniformly bounded. The constant is independent of $s=\epsilon$.

This proves the estimate for $z_1$, using higher initial norms where needed. The second equation is

Its source is $O(\epsilon )$ in every spatial Sobolev order. Terms with ${{\partial }_t}^2{z}_1$ are controlled by ${{\partial }_t}^2{z}_1=\Delta z_1$; all coefficient derivatives belong to the fixed smooth backgrounds. Equation (eq:wave-energy) proves the estimate for $z_2$.

Higher time derivatives follow inductively from these equations, using higher spatial norms and the fixed coefficient derivatives. Sobolev embedding gives spacetime smooth convergence away from $t=0$. Finally finite propagation removes the auxiliary spatial localization on the retained region.